1896.] On the Theory of the Partitions of Numbers. 197 



III. "Memoir on the Theory ot the Partitions of Numbers. 

 Part I." By Major P. A. MacMahon, R.A., F.R.S. 

 Received December 31, 1895. 



(Abstract.) 



The memoir here presented is a natural sequel to my memoirs of 

 1891, 1893, and 1894, published in extenso in the 1 Philosophical 

 Transactions.' In fundamental idea it is graphical, resting, on the 

 one hand, upon the method of the memoir on the " Compositions of 

 Numbers," of 1893, and, on the other, upon Sylvester's graphical 

 method, set forth in his " Constructive Theory of Partitions," of 

 1882, published in vol. 5 of the ' American Journal of Mathematics.' 



The memoir is divided into four sections. In § 1 I give new- 

 notions concerning the partitions of ordinary unipartite numbers, 

 and show that the theory of the separations of a partition necessitates 

 the consideration- of the partitions of multipartite numbers. The two 

 theories proceed in parallel paths. One-to-one correspondence can 

 be established at any point. 



In § 2 I am engaged with the graphical representation of unipar- 

 tite partitions. The graph that, in the memoir of 1893, was employed 

 to denote a principal composition of a bipartite number is shown to 

 be the graph also of a unipartite partition. A new theory of uni- 

 partite partitions is evolved with algebraical developments in cor- 

 respondence. 



In § 3 I investigate a similar correspondence between the composi- 

 tions of tripartite numbers and certain regularised partitions of 

 bipartite numbers. The method is of general application, and indi- 

 cates a one-to-one correspondence between the compositions of 

 ra+l-partite numbers and certain regularised partitions of w-partite 

 numbers. 



In § 4 I take up the question of the graphical representation of 

 completely regularised multipartite numbers. I follow Sylvester, 

 proceeding from two to three dimensions. Whereas Sylvester 

 employed nodes in a two-dimensional corner, 1 employ nodes piled 

 up in a three-dimensional corner. Sylvester obtains a two-fold 

 correspondence from the permutations of his axes x, y. I obtain a 

 six-fold correspondence from the permutations of the three axes 



y, z. Even Sylvester's two-dimensional graphs permit of six 

 interpretations when viewed from the three-dimensional standpoint. 



I strive to determine the number of graphs appertaining to a given 

 number of nodes, the numbers of nodes along the three axes being 

 restricted in any given manner. 



There is no difficulty informing a generating function which solves 



